課程名稱︰常微分方程導論 課程性質︰數學系大二必修 課程教師︰陳俊全教授 開課學院：理學院 開課系所︰數學系 考試日期︰2017年11月10日(五) 考試時限：08:10-09:00，共計50分鐘 試題 : 106 Introduction to Ordinary Differential Equations Quiz 1 Date: 2017.11.10 Write down all details in your argument as possible as you can. Part I: 1. (20%) Solve the following equations. (a) 2y' + xy = e^(-x^2/4). (b) y cos(x) + 2xe^y +(sin(x)+x^2e^y-1)y' = 0. 2. (20%) Solve the Bernoulli equation y' - 2y/x = -x^2y^2. 3. (20%) Solve (-3x+y+6)dx + (x+y+2)dy = 0. 4. (20%) Consider a pond that initially contains 10 million gal of fresh water. Water containing an undersirable chemical flows into the pond at the rate of 5 million gal/yr, and the mixture in the pond flows out at the same rate. The concentration γ(t) of chemical in the incoming water varies periodical- ly with time according to the expression γ(t) = 2 + sin(2t) g/gal. Determi- ne the amount of chemical in the pond at any time. Part II: Choose one of following two problems. 5. (20%) Recall the Picard iteration, which is a method of successive approxim- ations to approximate the solution of the initial value problem y' = f(t,y), (1) y(0) = y0, defined by ψ(t) = y0 and 0 t ψ(t) = y0 + ∫f(s,ψ(s))ds, n 屬於 |N (2) n 0 n-1 Consider f(t,y) = -(sin(t))y + 1 and y0 = 0. (a) Calculate ψ(t), ψ(t), ψ(t). 1 2 3 (b) Estimate the error: |ψ(t) - ψ(t)|, where ψ(t) is the solution to (1). 3 6. (20%) Consider the initial value problem y' = M^4 - y^4, (3) y(0) = y0, where M > 0 is a given constnant. Suppose 0 < y0 < M. Do not solve the equa- tion directly, use mathematical analysis to solve following problems. (a) Prove that y(t) is bounded and increasing. (b) Find lim y(t) and justify your result. t→∞ --

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